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\title{ALGORITHM OF NUMERICAL SIMULATION OF A PULSE
PROPAGATION IN NONUNIFORM MEDIUM}
\author{M.A.Antonets, L.V.Scherstneva}{
Radiophysical Research Institute, Nizhny Novgorod, Russia}

Solutions of initial value problem for hyperbolic system, generated
by wave equation, or for Maxwell system
$${\partial u\over \partial t}=A u,\quad u(0)=u_{0},$$
may be presented by Duhamel integral
$$u(t)=e^{t \hat{A}} u_{0}+\int_{0}^{t}{ e^{(t-s) \hat{A}} \vec{f}(s) ds}=
.$$
 Therefore for numerical simulation of an sound or electromagnetic pulse
propagation one needs effective algorithms for the operator exponent
$\exp{t \hat{A}}$ evaluation. For the purpose one may use basic
asymptotical relation
$$e^{t \hat{A}}=\widehat{e^{t A}}+o(t^2),\ t \to 0.$$
where  $\widehat{e^{t A}}$  is the operator corresponding to the Weyl symbol
$e^{t A}$ which is the matrix exponent of the Weyl matrix symbol $A$ of the
operator $\hat{A}$.
 In this case operator exponent  is the sum of regular and singular terms
$$e^{t \hat{A}}=\hat{U}(t)+\hat{K}(t)$$
where operator $\hat{U}(t)$ is similar to the operator exponent $\hat{A}$
in the case of the uniform medium, while  $\hat{K}(t)$ is an integral operator
and describes the scattering of the pulse on the medium inhomogeneities.
 It is possible, basing on this expansion, to construct an accurate
algorithm for  $\delta$-pulse scattering simulation. Since $$e^{t
\hat{A}} \delta(\vec r)=(\hat{U}(t/N)+\hat{K}(t/N))^{N}\delta+o(1/N)$$
then one can use the following successive procedure
$$(\hat{U}(t/N)+\hat{K}(t/N))u_{l-1}=u_l.$$
Where $u_l=u_l^r+u_l^s$, i.=A5. $u_l$ is a sum of the regular
$ u_l^r $ and the singular $u_l^s$ terms
$$u_l^r=\hat{U}(t/N)u_{l-1}^r+\hat{K}(t/N)u_{l-1}^s$$
$$u_l^s=\hat{U}(t/N)u_{l-1}^s. $$
Here the value $u_{N}^r$ describes the scattered part of wave field.

This work is supported by the  State High Education
Committee of Russian Federation, Grant 95-0-21-77.
\end{document}
